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A higher order analogue of the Carathéodory–Julia theorem. (English) Zbl 1102.30028
The classical Carathéodory-Julia theorem states that if \(w\) is in the Schur class \(S\) of analytic mappings from the open unit ball \(D\) into its closure and \(t_0\in T\) is a boundary point of \(D\), then the following are equivalent:
(1) \(d_1:= \liminf_{z \to t_0} \frac{1 -|w(z)|^2}{1 -|z|^2} < \infty.\)
(2) \(d_1:= \lim_{z {\widehat\to} t_0} \frac{1 -|w(z)|^2}{1 -|z|^2} < \infty.\)
(3) The limits \(w_0:= \lim_{z {\widehat\to} t_0} w(z)\) and \(d_3:= \lim_{z {\widehat\to} t_0}\frac{1 - w(z) \overline w_0}{1 - z \bar t_0}\) exist and satisfy \(|w_0| =1\) and \(d_3 \geq 0.\)
(4) The limits \(w_0 := \lim_{z {\widehat\to} t_0} w(z)\) and \(w_1 := \lim_{z {\widehat\to} t_0} w'(z)\) exist and satisfy \(|w_0| =1\) and \(t_0 w_1\overline w_0 \geq 0.\)
Here \(z {\widehat\to} t_0\) means that \(z\) approaches \(t_0\) non-tangentially. Moreover, when these conditions hold, \(d_1 = d_2 = d_3 =t_0 w_1\overline w_0.\)
In this paper the authors prove a higher order analogue of the Carathéodory-Julia theorem. In order to formulate this result some notation must be introduced. The Schwarz-Pick matrix of the Schur function \(w \in S\) is given by
\[ {\mathbb P}^w_n(z):= \biggl(\frac{1}{i!j!}\frac{\partial^{i+j}}{\partial z^i \partial \overline z^j}\frac{ 1 -|w(z)|^2}{1 -|z|^2}\biggr)_{i,j=0}^n. \] For a given boundary point \(t_0\in T\), the boundary Schwarz-Pick matrix is
\[ {\mathbb P}^w_n(t_0)= \lim_{z \widehat\to t_0} {\mathbb P}^w_n(z), \;\;(n\geq 0), \] provided this limit exists. Further assume that \(w \in S\) has non-tangential boundary limits \[ w_j(t_0):= \lim_{z \widehat\to t_0}\frac{w^{(j)}(z)}{j!}, \quad j=0,\dots,2n+1, \] and let \[ \mathbb P^w_n(t_0):= \begin{pmatrix} w_1(t_0) & \ldots & w_{n+1}(t_0) \\ \vdots & \;& \vdots \\ w_{n+1}(t_0) & \ldots & w_{2n+1}(t_0) \end{pmatrix} {\Psi}_n(t_0) \begin{pmatrix} \overline{w_0(t_0)} & \ldots & \overline {w_n(t_0)} \\ \;& \ddots & \vdots \\ 0 & \;& \overline{w_0(t_0)} \end{pmatrix}, \] where the first factor is a Hankel matrix, the third factor is an upper triangular Toeplitz matrix and \({ \Psi}_n(t_0) = ( \Psi_{jl})_{j,l=0}^n\) is the upper triangular matrix with entries \(\Psi_{jl} = 0\), if \(j> l\) and \(\Psi_{jl} = (-1)^l \binom{l}{j} t_0^{l+j+1}\), if \(j \leq l\). The lower right corner in the Schwarz-Pick matrix \({\mathbb P}^w_n(z)\) is denoted by \[ d_{w,n}(z):=\frac{1}{(n!)^2}\frac{\partial^{2n}}{\partial z^n \partial \bar z^n}\frac{ 1 -|w(z)|^2}{1 -|z|^2}. \] The main result can be stated as follows: Theorem. For \(w \in S\), \(t_0\in T\) and \(n \in Z_+\), the following are equivalent:
(1) \(\liminf_{z \to t_0} d_{w,n}(z) < \infty.\)
(2) \(\lim_{z {\widehat\to} t_0} d_{w,n}(z) < \infty.\)
(3) The boundary Schwarz-Pick matrix \({\mathbb P}^w_n(t_0)\) exists.
(4) The non-tangential boundary limits \(w_j(t_0)\) exists and satisfy \(|w_0(t_0)| =1\) and \(\mathbb P^w_n(t_0)\geq 0,\) where \(\mathbb P^w_n(t_0)\) is the matrix defined above.
Moreover, when these conditions hold, the limits above in (1) and (2) are equal and furthermore, \({\mathbb P}^w_n(t_0)=\mathbb P^w_n(t_0).\)
This last equality makes it possible to compute boundary Schwarz-Pick matrices in terms of boundary values of \(w\) and of its derivatives. This is in certain cases much easier to do than to use the original definition of \({\mathbb P}^w_n(t_0).\) On the other hand, the conditions in (4) impose some restrictions on the boundary limits \(w_j(t_0)\). When \(n=0\) this main result reduces to the classical Carathéodory-Julia theorem with statement (3) excluded.

30D40 Cluster sets, prime ends, boundary behavior
46E20 Hilbert spaces of continuous, differentiable or analytic functions
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
Full Text: DOI
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